College Uses Math Software to Interest Students in Calculus
Lafayette College is a small liberal-arts college in Easton, Pa., with an unusually large mathematics faculty. Over the past five years the math department has integrated a software program called Mathematica into its curriculum, beginning with its Scientific Calculus course. Several Lafayette faculty members teach a three-semester scientific and engineering calculus sequence, all using Mathematica-based teaching materials.
From Champaign, Ill.-based Wolfram Research, Mathematica can be characterized by its ability to perform symbolic calculations and extensive graphics, as well as the type of numerical computation commonly associated with computers.
New capabilities of software systems like Mathematica have a profound impact on the nature of mathematical investigation. The very existence of such tools necessitates a careful rethinking of the undergraduate mathematics curriculum.
For example, the value of having skills in calculational proficiency diminishes with the arrival of such tools. Instead, it becomes important that students learn the concepts underlying the calculations, so that they recognize opportunities to apply this powerful tool.
A New Look at Math
When using Mathematica, students no longer slog through long, difficult calculations, losing interest. Students can think more deeply about the concepts involved, and develop a better understanding of the ideas that underlie these calculations, especially the links with the geometry used to visualize them.
"Mathematica's graphics capabilities also offer a tremendous advantage," comments Dr. Robert Root, assistant professor of mathematics at Lafayette. "By alleviating the wearisome chore of graphing, the software leaves students with energy to look at what they have. This is especially evident in multivariable Calculus, where the 3D graphics can be particularly tedious and difficult to sketch accurately. Mathematica plots, in contrast, are simple to create and easy to interpret."
"I am convinced that what separates most students from the mathematical maturity discussed above is an ability to visualize what is going on," notes Root. "If I can show them an insightful plot or animation and make a connection with the algebra, they see calculus in a new way. This revelation is especially rewarding for students who never realized they had a talent for mathematics."
Concepts, Not Complications
Lafayette calculus students attend three lectures and one lab session a week, each lasting 75 minutes. Root relies on Mathematica's cross-platform compatibility to create his courseware on a Macintosh, then ports it to Windows for the lab's PCs.
In the lab, students use interactive Mathematica files, called "notebooks," plus lab handouts, which they complete with an accompanying lab report. Because the software is also a programming language, Root's lab notebooks often combine standard Mathematica with a few programmed functions that illustrate particular points.
He has written one such package that animates Newton's method. Another illustrates the "Brachistochrone Problem" with a race between beads sliding down different curves. These visualizations help the students to focus on the concepts, and sustain their interest through the accompanying algebra and calculus.
Traditional vs. Progressive
Root and the other professors must consider carefully how the Mathematica labs affect the rest of the course and how students think about math. "It's quite a shock for many first-year students to discover their assignments exclude problems with long, involved computations. That is no longer an important part of the curriculum," says Root.
Even in the traditional areas of the course, the presence of Mathematica is pervasive. It affects every lecture by Root and his colleagues. The use is deliberate ¬ to change the students' outlook on mathematics -- on what they can do, and on what is reasonable for them to attempt.
"I want students who not only understand the math concepts involved, but who can also perform perfunctory calculations without Mathematica," Root says. "I expect them to turn to the program only for intricate problems, to avoid wasting time on extensive calculations."
"We cannot reach all of our students," admits Root. "I nonetheless believe that Mathematica helps reach more, and in particular helps us encourage students who have never been excited about mathematics to develop their analytical abilities."